Proposition 7.3.4.label Let $X$ be a compact topological space and $E$ be a TVS over $K \in \RC$, then:

1. (1)

   $C(X; E)$ equipped with the uniform topology is a topological vector space over $K$.

2. (2)

   If $E$ is locally convex and $\rho: E \to [0, \infty)$ is a continuous seminorm, then

   \[[\cdot]_{u, \rho}: C(X; E) \to [0, \infty) \quad f \mapsto \sup_{x \in X}\rho(f(x))\]

   
   

   is a continuous seminorm on $C(X; E)$ with respect to the uniform topology.

   The uniform topology on $C(X; E)$ is locally convex, and induced by seminorms of the form $[\cdot]_{u, \rho}$, where $\rho$ ranges over all continuous seminorms on $E$.

3. (3)

   If $E$ is normed, then

   \[[\cdot]_{u}: C(X; E) \quad f \mapsto \sup_{x \in X}\norm{f(x)}_{E}\]

   
   

   is a norm on $C(X; E)$ that induces the uniform topology.

4. (4)

   If $E$ is complete, then so is $C(X; E)$.

As such, the uniform topology is the canonical topology on $C(X; E)$.